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We obtain a list of simple classes of singularities of function germs with respect to the quasi m-boundary equivalence relation, with m ≥ 2 . The results obtained in this paper are a natural extension of Zakalyukin's work on the new non-standard equivalent relation. In spite of the rather artificial nature of the definitions, the quasi relations have very natural applications in symplectic geometry. In particular, they are used to classify singularities of Lagrangian projections equipped with a submanifold. The main method that is used in the classification is the standard Moser's homotopy technique. In addition, we adopt the version of Arnold's spectral sequence method, which is described in Lemma 2. Our main results are Theorem 4 on the classification of simple quasi classes, and Theorem 5 on the classification of Lagrangian submanifolds with smooth varieties. The brief description of the main results is given in the next section. [ABSTRACT FROM AUTHOR]